On Basic Fourier-Bessel Expansions

Abstract

When dealing with Fourier expansions using the third Jackson (also known as Hahn-Exton) $q$-Bessel function, the corresponding positive zeros $j_{k\nu }$ and the "shifted" zeros, $q{j}_{k\nu }$, among others, play an essential role. Mixing classical analysis with $q$-analysis we were able to prove asymptotic relations between those zeros and the "shifted" ones, as well as the asymptotic behavior of the third Jackson $q$-Bessel function when computed on the "shifted" zeros. A version of a $q$-analogue of the Riemann-Lebesgue theorem within the scope of basic Fourier-Bessel expansions is also exhibited.

References in corpus (3)

• Completeness, special functions and uncertainty principles over q-linear grids
• The Nevanlinna parametrization for $q$-Lommel polynomials in the indeterminate case
• Nevanlinna extremal measures for polynomials related to $q^{-1}$-Fibonacci polynomials

Cited by in corpus (2)

• Uniform convergence of Fourier-Bessel series on a q-linear grid
• A $\beta$-Sturm Liouville problem associated with the general quantum operator